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Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product."

I try my best to understand Stone-Cech compactification theorem by a not obviously-trivial example. However, when I consider the exponential map in complex plane, I fail to imagine the behaviour for exponential map in compactification space. I guess the compactification space does have complex structure. Can we say more about the further knowledge about the compactification space and the exponential maps in it.

Any comments and remarks will be appreciated.

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  • $\begingroup$ What does this question have to do with the tag 'differential-operators'? $\endgroup$
    – Ricardo Andrade
    Commented Oct 26, 2013 at 13:39

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Convergence in the Stone-Cech compactification is equivalent to convergence of values of any bounded continuous function, and there is really a lot of them. In particular, geodesics in symmetric spaces do not converge in the S-C compactification.

However, I don't think that the S-C compactification has any relevance to products of matrices. There are much more geometrical compactifications and boundaries of the associated symmetric spaces.

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    $\begingroup$ Hmm..... (at least the part about "bounded" connects to the question fine). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Oct 9, 2013 at 0:12
  • $\begingroup$ @Wlodzimierz Holsztynski - ??? $\endgroup$
    – R W
    Commented Oct 9, 2013 at 1:34
  • $\begingroup$ There is not much of a question. And there is not much of an answer. One could turn this question into something focused: present examples of a combination of a topological space $X$ and an additional related structure on it which would induce some related structure on $\beta(X)$ (or perhaps on $\beta(X)\setminus X$). Topological algebras would be important. But also other structures and their combinations. That's how I see it. (Of course the $\exp$ applies rather to bounded subsets of $\mathbb C$). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Oct 9, 2013 at 3:00

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